Quiz — Serie d'exercices Corrigés - Math - Suites réelles 1 - 4ème Math (2009-2010)_2.pdf

Question 1 sur 10 20:00

[{"id":14254,"question":"Soit une suite (uₙ) définie par uₙ = 3n² - 5n + 2. Que peut-on dire de sa monotonie ?","option_a":"Croissante à partir d'un certain rang","option_b":"Décroissante à partir d'un certain rang","option_c":"Constante","option_d":"Non monotone","option_e":"","option_f":"","bonne_reponse":"a","explication":"La suite est croissante à partir d'un certain rang car le terme dominant 3n² domine pour n grand, et la dérivée (6n - 5) est positive pour n ≥ 1.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"a\", \"options\": {\"a\": \"Croissante à partir d'un certain rang\", \"b\": \"Décroissante à p","_debug_options_count":4},{"id":14255,"question":"La suite (vₙ) définie par vₙ = (-1)ⁿ est-elle convergente ?","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"b","explication":"La suite alterne entre -1 et 1, elle n'a pas de limite finie, donc elle est divergente.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"b\", \"options\": {\"a\": \"Vrai\", \"b\": \"Faux\", \"c\": \"\", \"d\": \"\"}}","_debug_options_count":4},{"id":14256,"question":"Quelle est la limite de la suite (wₙ) définie par wₙ = (2n + 1)\/(n - 3) quand n tend vers l'infini ?","option_a":"2","option_b":"0","option_c":"1\/2","option_d":"∞","option_e":"","option_f":"","bonne_reponse":"a","explication":"En divisant numérateur et dénominateur par n, on obtient (2 + 1\/n)\/(1 - 3\/n) → 2\/1 = 2 quand n → ∞.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"a\", \"options\": {\"a\": \"2\", \"b\": \"0\", \"c\": \"1\/2\", \"d\": \"∞\"}}","_debug_options_count":4},{"id":14257,"question":"Soit (uₙ) une suite croissante et majorée. Que peut-on conclure ?","option_a":"Elle est convergente","option_b":"Elle est divergente","option_c":"Elle est constante","option_d":"On ne peut rien dire","option_e":"","option_f":"","bonne_reponse":"a","explication":"D'après le théorème de la limite monotone, une suite croissante et majorée est convergente.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"a\", \"options\": {\"a\": \"Elle est convergente\", \"b\": \"Elle est divergente\", \"c\": \"Elle est","_debug_options_count":4},{"id":14258,"question":"La suite (uₙ) définie par u₀ = 1 et uₙ₊₁ = √(2 + uₙ) est-elle croissante ?","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"En calculant les premiers termes, on observe u₁ ≈ 1.732 \u003E u₀ = 1, et par récurrence, la suite est croissante.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"a\", \"options\": {\"a\": \"Vrai\", \"b\": \"Faux\", \"c\": \"\", \"d\": \"\"}}","_debug_options_count":4},{"id":14259,"question":"Quelle est la limite de la suite (uₙ) définie par uₙ = n\/(n + 1) ?","option_a":"0","option_b":"1","option_c":"∞","option_d":"1\/2","option_e":"","option_f":"","bonne_reponse":"b","explication":"En divisant numérateur et dénominateur par n, on obtient 1\/(1 + 1\/n) → 1 quand n → ∞.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"b\", \"options\": {\"a\": \"0\", \"b\": \"1\", \"c\": \"∞\", \"d\": \"1\/2\"}}","_debug_options_count":4},{"id":14260,"question":"Soit (uₙ) une suite telle que uₙ ≤ vₙ ≤ wₙ pour tout n, et que lim uₙ = lim wₙ = L. Que peut-on dire de la suite (vₙ) ?","option_a":"Elle converge vers L","option_b":"Elle diverge","option_c":"On ne peut rien dire","option_d":"Elle est constante","option_e":"","option_f":"","bonne_reponse":"a","explication":"D'après le théorème des gendarmes, si uₙ ≤ vₙ ≤ wₙ et lim uₙ = lim wₙ = L, alors lim vₙ = L.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"a\", \"options\": {\"a\": \"Elle converge vers L\", \"b\": \"Elle diverge\", \"c\": \"On ne peut rien","_debug_options_count":4},{"id":14261,"question":"La suite (uₙ) définie par uₙ = (-2)ⁿ est-elle bornée ?","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"b","explication":"La suite alterne entre des valeurs positives et négatives de plus en plus grandes en valeur absolue, donc elle n'est pas bornée.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"b\", \"options\": {\"a\": \"Vrai\", \"b\": \"Faux\", \"c\": \"\", \"d\": \"\"}}","_debug_options_count":4},{"id":14262,"question":"Quelle est la limite de la suite (uₙ) définie par uₙ = (n² + 3n)\/(2n² - 5) ?","option_a":"1\/2","option_b":"0","option_c":"∞","option_d":"3\/2","option_e":"","option_f":"","bonne_reponse":"a","explication":"En divisant numérateur et dénominateur par n², on obtient (1 + 3\/n)\/(2 - 5\/n²) → 1\/2 quand n → ∞.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"a\", \"options\": {\"a\": \"1\/2\", \"b\": \"0\", \"c\": \"∞\", \"d\": \"3\/2\"}}","_debug_options_count":4},{"id":14263,"question":"Soit (uₙ) une suite telle que uₙ₊₁ = uₙ² - 2uₙ + 2. Si u₀ = 2, que vaut u₂ ?","option_a":"2","option_b":"1","option_c":"0","option_d":"4","option_e":"","option_f":"","bonne_reponse":"b","explication":"u₁ = 2² - 2*2 + 2 = 2, u₂ = 2² - 2*2 + 2 = 2 - 4 + 2 = 1. La suite converge vers 1.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"b\", \"options\": {\"a\": \"2\", \"b\": \"1\", \"c\": \"0\", \"d\": \"4\"}}","_debug_options_count":4}]